where \(U\sim \text{Haar}(m)\)  and \(V\sim \text{Haar}(n)\)  are two independent random orthogonal matrices, and \(\Sigma \in \mathbb {R}^{m \times n}\)  is a rectangular diagonal matrix independent of \(U, V\) . Matrices like these often appear in the study of free probability theory [1]}. They are also used as a convenient model for matrices whose singular vectors are generic [2]}, [3]}, [4]}. Strictly speaking, Theorem REF  only applies to the case where the dynamics operates on a single random orthogonal matrix. However, it is obvious from the proof that the idea applies to more general dynamics involving a finite number of independent random orthogonal matrices. Thus, Algorithm REF  can be easily adapted to handle the matrix ensemble given in (REF ).
